Nonanalytic Points Research Articles

Stretched-exponential relaxation in weakly confined Brownian systems through large deviation theory.

Stretched-exponential relaxation is a widely observed phenomenon found in ordered ferromagnets as well as glassy systems. One modeling approach connects this behavior to a droplet dynamics described by an effective Langevin equationfor the droplet radius with an r^{2/3} potential. Here, we study a Brownian particle under the influence of a general confining, albeit weak, potential field that grows with distance as a sublinear power law. We find that for this memoryless model, observables display stretched-exponential relaxation. The probability density function of the system is studied using a rate-function ansatz. We obtain analytically the stretched-exponential exponent along with an anomalous power-law scaling of length with time. The rate function exhibits a point of nonanalyticity, indicating a dynamical phase transition. In particular, the rate function is double valued both to the left and right of this point, leading to four different rate functions, depending on the choice of initial conditions and symmetry.

Mechanism of dynamical phase transitions: The complex-time survival amplitude

Dynamical phase transitions are defined through nonanalyticities of the survival probability of an out-of-equilibrium time-evolving state at certain critical times. They ensue from zeros of the corresponding survival amplitude. By extending the time variable onto the complex domain, we formulate the complex-time survival amplitude. The complex zeros of this quantity near the time axis correspond, in the infinite-size limit, to nonanalytical points where the survival probability abruptly vanishes. Our results are numerically exemplified in the fully connected transverse-field Ising model, which displays a symmetry-broken phase delimited by an excited-state quantum phase transition. A detailed study of the behavior of the complex-time survival amplitude when the characteristics of the out-of-equilibrium protocol change is presented. The influence of the excited-state quantum phase transition is also put into context.

Renormalisation of non-differentiable potentials

Non-differentiable potentials, such as the V-shaped (linear) potential, appear in various areas of physics. For example, the effective action for branons in the framework of the brane world scenario contains a Liouville-type interaction, i.e., an exponential of the V-shaped function. Another example is coming from particle physics when the standard model Higgs potential is replaced by a periodic self-interaction of an N-component scalar field which depends on the length, thus it is O(N) symmetric. We first compare classical and quantum dynamics near non-analytic points and discuss in this context the role of quantum fluctuations. We then study the renormalisation of such potentials, focusing on the Exact Wilsonian Renormalisation approach, and we discuss how quantum fluctuations smoothen the bare singularity of the potential. Applications of these results to the non-differentiable effective branon potential and to the O(N) models when the spatial dimension is varied and to the O(N) extension of the sine-Gordon model in (1+1) dimensions are presented.

Entanglement and precession in two-dimensional dynamical quantum phase transitions

Nonanalytic points in the return probability of a quantum state as a function of time, known as dynamical quantum phase transitions (DQPTs), have received great attention in recent years, but the understanding of their mechanism is still incomplete. In our recent work [Phys. Rev. Lett. 126, 040602 (2021)], we demonstrated that one-dimensional DQPTs can be produced by two distinct mechanisms, namely semiclassical precession and entanglement generation, leading to the definition of precession (pDQPTs) and entanglement (eDQPTs) dynamical quantum phase transitions. In this manuscript, we extend and investigate the notion of p- and eDQPTs in two-dimensional systems by considering semi-infinite ladders of varying width. For square lattices, we find that pDQPTs and eDQPTs persist and are characterized by similar phenomenology as in 1D: pDQPTs are associated with a magnetization sign change and a wide entanglement gap, while eDQPTs correspond to suppressed local observables and avoided crossings in the entanglement spectrum. However, DQPTs show higher sensitivity to the ladder width and other details, challenging the extrapolation to the thermodynamic limit especially for eDQPTs. Moving to honeycomb lattices, we also demonstrate that lattices with an odd number of nearest neighbors give rise to phenomenologies beyond the one-dimensional classification.

Mu PT statistical ensemble: systems with fluctuating energy, particle number, and volume

Within the theory of statistical ensemble, the so-called mu PT ensemble describes equilibrium systems that exchange energy, particles, and volume with the surrounding. General, model-independent features of volume and particle number statistics are derived. Non-analytic points of the partition function are discussed in connection with divergent fluctuations and ensemble equivalence. Quantum and classical ideal gases, and a model of Bose gas with mean-field interactions are discussed as examples of the above considerations.

Nonanalyticity, sign problem, and Polyakov line in a Z3 -symmetric heavy quark model at low temperature: Phenomenological model analyses

The nonanalyticity and the sign problem in the Z3-symmetric heavy quark model at low temperature are studied phenomenologically. For the free heavy quarks, the nonanalyticity is analyzed in the relation to the zeros of the grand canonical partition function. The Z3-symmetric effective Polyakov-line model (EPLM) in strong coupling limit is also considered as an phenomenological model of Z3-symmetric QCD with large quark mass at low temperature. We examine how the Z3-symmetric EPLM approaches to the original one in the zero-temperature limit. The effects of the Z3-symmetry affect the structure of zeros of the microscopic probability density function at the nonanalytic point. The average value of the Polyakov line can detect the structure, while the other thermodynamic quantities are not sensible to the structure in the zero-temperature limit. The effect of the imaginary quark chemical potential is also discussed. The imaginary part of the quark number density is very sensitive to the symmetry structure at the nonanalytical point. For a particular value of the imaginary quark number chemical potential, large quark number may be induced in the vicinity of the nonanalytical point.

Proliferation of non-linear excitations in the piecewise-linear perceptron

We investigate the properties of local minima of the energy landscape of a continuous non-convex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and non-linear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres. The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two non-analytic points where the derivative has a jump. Correspondingly, in the non-convex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of non-linear excitations as the number of non-analytic points in the cost function is increased.

Extreme value theory for constrained physical systems.

We investigate extreme value theory for physical systems with a global conservation law which describes renewal processes, mass transport models, and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support. We expose exact relationships between constrained extreme value theory and well-known quantities of the underlying stochastic dynamics, all valid beyond the midpoint in general, i.e., even far from the thermodynamic limit. For example, for renewal processes the distribution of the maximum time between two renewal events is exactly related to the mean number of these events. In the thermodynamic limit, we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory. For example, for the renewal process we unravel dual scaling of the extreme value distribution, pointing out two types of limiting laws: a normalizable scaling function for the typical statistics and a non-normalized state describing the rare events.

The edge states properties of the topological insulator with fermion path nonanalyticity points located at the edge

The edge states properties of a finite-size 2D topological insulator (TI) with the open boundary conditions in both directions are studied. It is shown that fermion path nonanalyticity points appearing on the edge of finite-size TI cause a nonuniform edge state distribution along the TI edge. The character of this distribution depends substantially on the edge state’s energy position in the bulk band gap. The edge state with the energy in the middle of the gap tends to be located in the corners between two nonparallel edges of the system, while the edge states with the energy close to the bulk band edge avoid the corners. The monotonic transition from one space distribution of the edge state wave function to another along the edge state’s band is observed. The mentioned edge state’s structure leads to the nonlinear current-voltage characteristic of the square-shaped TI with contacts connected with the corners of the device. It opens the possibility to control the current with the gate voltage and is useful for construction the nano-size devices. For the edge states with energy in the middle of a bulk gap, the vortex structure of the probability flux located near corners is found.

Signatures of quantum phase transitions from the boundary of the numerical range

The ground state energy of a finite-dimensional one-parameter Hamiltonian and the continuity of a maximum-entropy inference map are discussed in the context of quantum critical phenomena. The domain of the inference map is a convex compact set in the plane, called the numerical range. We study the differential geometry of its boundary in relation to the ground state energy. We prove that discontinuities of the inference map correspond to $C^1$-smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at $C^1$-smooth points of the boundary of the numerical range considered as a manifold. Discontinuities exist at all $C^2$-smooth non-analytic boundary points and are essentially stronger than at analytic points or at points which are merely $C^1$-smooth (non-exposed points).

Singularities in large deviations of work in quantum quenches

We investigate large deviations of the work performed in a quantum quench across two different phases separated by a quantum critical point, using as example the Dicke model quenched from its superradiant to its normal phase. We extract the distribution of the work from the Loschmidt amplitude and compute for both the corresponding large-deviation forms. Comparing these findings with the predictions of the classification scheme put forward in [Phys. Rev. Lett. 109, 250602 (2012)], we are able to identify a regime which is in fact distinct to the ones identified so far: here the rate function exhibits a non-analytical point which is a strong indication of the existence of an out-of-equilibrium phase transition in the rare fluctuations of the work.

Non-analytic behavior of the Loschmidt echo in XXZ spin chains: Exact results

We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise from the level crossings of this matrix. Our exact results are expressed in terms of “excited-state” thermodynamic Bethe ansatz equations, whose solutions involve non-trivial Riemann surfaces. By evaluating our formulas, we provide explicit numerical results for the quench from the Néel state, and we determine the first few non-analytic points.

Effect of Total Reflection from a Symmetric Two-Channel Device with Fermion Path Nonanalyticity Points Induced by Rashba Spin-Orbit Coupling

The effect Rashba spin-orbit coupling has on transmission coefficient through a symmetric system with fermion path nonanalyticity points is illustrated using the example of a regular polygon-shaped chain. It is shown that the current passage through a device is blocked at the critical spin-orbit coupling values determined by the system’s geometry. At the near-critical spin-orbit coupling values, electron transport is possible only in a narrow range of energies.

Geometrical interpretation of dynamical phase transitions in boundary-driven systems.

Dynamical phase transitions are defined as nonanalytic points of the large deviation function of current fluctuations. We show that for boundary-driven systems, many dynamical phase transitions can be identified using the geometrical structure of an effective potential of a Hamiltonian, recovered from the macroscopic fluctuation theory description. Using this method we identify new dynamical phase transitions that could not be recovered using existing perturbative methods. Moreover, using the Hamiltonian picture, an experimental scheme is suggested to demonstrate an analog of dynamical phase transitions in linear, rather than exponential, time.

Causality and information transfer in simultaneously slow- and fast-light media.

We demonstrate the simultaneous propagation of slow- and fast-light optical pulses in a four-wave mixing scheme using warm potassium vapor. We show that when the system is tuned such that the input probe pulses exhibit slow-light group velocities and the generated pulses propagate with negative group velocities, the information velocity in the medium is nonetheless constrained to propagate at, or less than, c. These results demonstrate that the transfer and copying of information on optical pulses to those with negative group velocities obeys information causality, in a manner that is reminiscent of a classical version of the no-cloning theorem. Additionally, these results support the fundamental concept that points of non-analyticity on optical pulses correspond to carriers of new information.

Fano antiresonance induced by the rashba spin−orbit coupling in systems with conduction channels exhibiting the points of nonanalyticity in fermion paths

It is shown that the combined effect of the nonanalyticity of the channel for the motion of charge carriers and the Rashba spin−orbit coupling induces resonant anomalies in the transport characteristics of nanosystems related to the size quantization. When the characteristic length determined by the ratio of the hopping integral and the spin−orbit coupling constant coincides with the distance between the points of nonanalyticity, the size effect arises in the channel. It manifests itself in the complete reflection from the device, which can be treated as the Fano antiresonance. The current−voltage characteristics of the nanosystem with the nonanalytical channel undergo significant changes at slight variations of the spin−orbit coupling constant near its critical value.

Propagation of phase nonanalytical points in fast- and slow-light media

Propagation of phase nonanalytical points in fast- and slow-light media

A minimal model of dynamical phase transition

We calculate the large deviation functions characterizing the long-time fluctuations of the occupation of drifted Brownian motion and show that these functions have non-analytic points. This provides the first example of dynamical phase transition that appears in a simple, homogeneous Markov process without an additional low-noise, large-volume or hydrodynamic scaling limit.

Blockage-induced condensation controlled by a local reaction.

We consider the setup of stationary zero range models and discuss the onset of condensation induced by a local blockage on the lattice. We show that the introduction of a local feedback on the hopping rates allows us to control the particle fraction in the condensed phase. This phenomenon results in a current versus blockage parameter curve characterized by two nonanalyticity points.

Nonanalytic pulse discontinuities as carriers of information

In this paper, we consider optical pulses encoded with two nonanalytic points, and we evaluate the detectable information of these pulses in media supporting slow- and fast-light propagation. It is shown that, in some configurations of slow- (subluminal) light propagation, the signal is not readily detectable, albeit the arrival of the encoded nonanalytic points at the receiver. It is thus argued that, from a practical point of view, information should not be entirely associated with the pulse discontinuities. On the other hand, it is confirmed that for propagation in vacuum or a fast-light medium, detectable information is bounded by the nonanalytic points, which create a space-time window within which detectable information cannot escape. As such, the distinction between the nonanalytic points of a signal, its energy transport, and detectable information is demonstrated.